用可证明的数据增强提升求解二次规划的神经网络泛化能力
Principled Data Augmentation for Learning to Solve Quadratic Programming Problems
- 基于理论保证生成保持最优性的数据，增强训练多样性
- 在少量标注数据下仍能显著提升模型对二次规划的求解准确率
- 适合需要高效求解复杂优化问题的研究者与工程应用
线性与二次优化在机器学习训练、整数线性规划等众多实际场景中至关重要。近年来，利用消息传递图神经网络（MPNN）的“学习求解”方法（L2O）在求解线性规划（LP）或二次规划（QPs）方面日益流行，其作为轻量级、数据驱动的替代方案，可显著降低分支定界求解器中强分支评分的计算开销。然而，在数据稀缺环境下，鲁棒的L2O MPNN仍面临挑战，尤其在处理复杂二次规划问题时。本文提出一种针对二次规划的原理性数据增强方法，通过理论支持的数据增强技术生成多样化且保持最优性的实例。进一步地，将这些增强数据融入自监督对比学习框架，实现对MPNN的预训练，从而提升其在各类学习求解任务中的表现。大量实验表明，该方法在监督场景中增强泛化能力，并有效支持向相关优化问题的迁移学习。
原文摘要 · Abstract (English)
Linear and quadratic optimization are crucial in numerous real-world applications, ranging from training machine learning models to solving integer linear programs. Recently, learning-to-optimize methods (L2O) for linear (LPs) or quadratic programs (QPs) using message-passing graph neural networks (MPNNs) have gained traction, promising lightweight, data-driven proxies for solving such optimization problems. For example, they replace the costly computation of strong branching scores in branch-and-bound solvers, thereby reducing the need to solve many such optimization problems. However, robust L2O MPNNs remain challenging in data-scarce settings, especially when addressing complex optimization problems such as QPs. This work introduces a principled approach to data augmentation tailored for QPs via MPNNs. Our method leverages theoretically justified data augmentation techniques to generate diverse yet optimality-preserving instances. Furthermore, we integrate these augmentations into a self-supervised contrastive learning framework, thereby pretraining MPNNs for improved performance on L2O tasks. Extensive experiments demonstrate that our approach improves generalization in supervised scenarios and facilitates effective transfer learning to related optimization problems.
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